A tighter upper bound for random MAX 2-SAT

Abstract Given a conjunctive normal form F with n variables and m = c n 2-clauses, it is interesting to study the maximum number max F of clauses satisfied by all the assignments of the variables (MAX 2-SAT). When c is sufficiently large, the upper bound of f ( n , c n ) ≐ E ( max F ) of random MAX 2-SAT had been derived by the first-moment argument. In this paper, we provide a tighter upper bound ( 3 / 4 ) c n + g ( c ) c n also using the first-moment argument but correcting the error items for f ( n , c n ) , and g ( c ) = ( 3 / 4 ) cos ( ( 1 / 3 ) × arccos ( ( 4 ln 2 ) / c − 1 ) ) − 3 / 8 when considering the e 3 error item. Furthermore, we extrapolate the region of the validity of the first-moment method is c > 2.4094 by analyzing the higher order error items.